# Definition:Peano Structure

## Definition

A **Peano structure** $\struct {P, 0, s}$ comprises a set $P$ with a successor mapping $s: P \to P$ and a non-successor element $0$.

These three together are required to satisfy Peano's axioms.

## Also known as

A **Peano structure** is also known as a **Dedekind-Peano structure**.

## Source of Name

This entry was named for Giuseppe Peano and Richard Dedekind.

## Historical Note

A set of axioms on the same topic as **Peano's axioms** was initially formulated by Richard Dedekind in $1888$.

Giuseppe Peano published them in $1889$ according to his own formulation, in a more precisely stated form than Dedekind's.

Bertrand Russell pointed out that while **Peano's axioms** give the key properties of the natural numbers, they do not actually define what the natural numbers actually are.

According to 1960: Paul R. Halmos: *Naive Set Theory*:

*[These] assertions ... are known as the Peano axioms; they used to be considered as the fountainhead of all mathematical knowledge.*

It is worth pointing out that the **Peano axioms** can be deduced to hold for the minimally inductive set as defined by the Axiom of Infinity from the Zermelo-Fraenkel axioms.

Thus they are now rarely considered as axiomatic as such.

However, in their time they were groundbreaking.